半线性heat equations零控制的最小二乘逼近
Approximation of null controls for semilinear heat equations using a least-squares approach
摘要：
半线性热方程$y_t-Δy + g(y)=f \,1_ω$的零分布能控性，假设$g$满足增长条件$g(s)/(\vert s\vert \log^{3/2}(1+\vert s\vert))\rightarrow 0$作为$\vert s\vert \rightarrow \infty$，并且$g^\prime\in L^\infty_{loc}(\mathbb{R})$已由Fernánal-Cara和Zuazua在2000年得到.基于不动点论证的证明利用了线性化热方程的可观测常数的精确估计。但是，它没有提供空控件的显式构造。假设对于一个$s\in (0,1]$，有$g^\prime\in W^{s,\infty}(\mathbb{R})$，我们构造了一个显式序列，该序列强收敛于零控制。该方法基于最小二乘法，推广了牛顿型方法，并保证了收敛性，无论是序列的初始元素。特别地，在有限次迭代之后，收敛是超线性的，其速率等于$1+s$。一维数值实验支持了我们的分析。
英文摘要：
The null distributed controllability of the semilinear heat equation $y_t-Δy + g(y)=f \,1_ω$, assuming that $g$ satisfies the growth condition $g(s)/(\vert s\vert \log^{3/2}(1+\vert s\vert))\rightarrow 0$ as $\vert s\vert \rightarrow \infty$ and that $g^\prime\in L^\infty_{loc}(\mathbb{R})$ has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a fixed point argument makes use of precise estimates of the observability constant for a linearized heat equation. It does not provide however an explicit construction of a null control. Assuming that $g^\prime\in W^{s,\infty}(\mathbb{R})$ for one $s\in (0,1]$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach, generalizes Newton type methods and guarantees the convergence whatever be the initial element of the sequence. In particular, after a finite number of iterations, the convergence is super linear with a rate equal to $1+s$. Numerical experiments in the one dimensional setting support our analysis.
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